Medical treatment of a patient often requires the use of a plurality of invasive lines which may be of differing lengths and which may be connected with the patient at a variety of body entry sites. Because these lines are flexible and may be dispersed throughout an are where the patient is being treated (e.g. a hospital bed or stretcher), they are subject to becoming tangled and disconnected from the patient and attendant medical equipment. Conventionally used schemes to deal with this problem (what is often referred to by hospital personnel as a `spaghetti` mess) have included taping the lines to a board or the patient's bed, pinning the lines to the bed, and clipping the lines to hemostats to be clipped to the patient, none of which has proven satisfactory. A more complicated and still unacceptable proposal to organize and identify invasive lines is described in U.S. Pat. No. 4,988,062 to R. A. London and involves the use of a multi-slotted manifold that is configured to be mounted upon a patient support surface, such as the patient's bed, and held in place or confined beneath the patient's headrest. Such a manifold structure merely provides a fixed bed-to-equipment interface from which multiple invasive lines fan out to the patient, on the one hand, and to attendant medical equipment external to the bed, on the other hand. Elsewhere, on and around the patient, the invasive lines are still subject to becoming kinked, snagged, tangled and disconnected from the patient, particularly if the patient is moved (e.g. rolled over).